1.
Mathematical physics
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Mathematical physics refers to development of mathematical methods for application to problems in physics. It is a branch of applied mathematics, but deals with physical problems, there are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The rigorous, abstract and advanced re-formulation of Newtonian mechanics adopting the Lagrangian mechanics, both formulations are embodied in analytical mechanics. These approaches and ideas can be and, in fact, have extended to other areas of physics as statistical mechanics, continuum mechanics, classical field theory. Moreover, they have provided several examples and basic ideas in differential geometry, the theory of partial differential equations are perhaps most closely associated with mathematical physics. These were developed intensively from the half of the eighteenth century until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics. The theory of atomic spectra developed almost concurrently with the fields of linear algebra. Nonrelativistic quantum mechanics includes Schrödinger operators, and it has connections to atomic, Quantum information theory is another subspecialty. The special and general theories of relativity require a different type of mathematics. This was group theory, which played an important role in quantum field theory and differential geometry. This was, however, gradually supplemented by topology and functional analysis in the description of cosmological as well as quantum field theory phenomena. In this area both homological algebra and category theory are important nowadays, statistical mechanics forms a separate field, which includes the theory of phase transitions. It relies upon the Hamiltonian mechanics and it is related with the more mathematical ergodic theory. There are increasing interactions between combinatorics and physics, in statistical physics. The usage of the mathematical physics is sometimes idiosyncratic. Certain parts of mathematics that arose from the development of physics are not, in fact, considered parts of mathematical physics. The term mathematical physics is sometimes used to research aimed at studying and solving problems inspired by physics or thought experiments within a mathematically rigorous framework
2.
Modern physics
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Modern physics is the post-Newtonian conception of physics. In general, the term is used to refer to any branch of physics either developed in the early 20th century and onwards, small velocities and large distances is usually the realm of classical physics. In general, quantum and relativist effects exist across all scales, in a literal sense, the term modern physics, means up-to-date physics. In this sense, a significant portion of so-called classical physics is modern, however, since roughly 1890, new discoveries have caused significant paradigm shifts, the advent of quantum mechanics and of Einsteinian relativity. Physics that incorporates elements of either QM or ER is said to be modern physics and it is in this latter sense that the term is generally used. Modern physics is often encountered when dealing with extreme conditions, quantum mechanical effects tend to appear when dealing with lows, while relativistic effects tend to appear when dealing with highs, the middles being classical behaviour. For example, when analysing the behaviour of a gas at room temperature, however near absolute zero, the Maxwell–Boltzmann distribution fails to account for the observed behaviour of the gas, and the Fermi–Dirac or Bose–Einstein distributions have to be used instead. Very often, it is possible to find – or retrieve – the classical behaviour from the description by analysing the modern description at low speeds. When doing so, the result is called the classical limit
3.
General relativity
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General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins theory has important astrophysical implications, for example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of an expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a thought experiment involving an observer in free fall. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, the Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory, but as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, in line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that our universe is expanding and this is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life
4.
Spacetime
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In physics, spacetime is any mathematical model that combines space and time into a single interwoven continuum. Until the turn of the 20th century, the assumption had been that the 3D geometry of the universe was distinct from time, Einsteins theory was framed in terms of kinematics, and showed how measurements of space and time varied for observers in different reference frames. His theory was an advance over Lorentzs 1904 theory of electromagnetic phenomena. A key feature of this interpretation is the definition of an interval that combines distance. Although measurements of distance and time between events differ among observers, the interval is independent of the inertial frame of reference in which they are recorded. The resultant spacetime came to be known as Minkowski space, non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space and which is separate from space. Classical mechanics assumes that time has a constant rate of passage that is independent of the state of motion of an observer, furthermore, it assumes that space is Euclidean, which is to say, it assumes that space follows the geometry of common sense. General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field. Mathematically, spacetime is a manifold, which is to say, by analogy, at small enough scales, a globe appears flat. An extremely large scale factor, c relates distances measured in space with distances measured in time, waves implied the existence of a medium which waved, but attempts to measure the properties of the hypothetical luminiferous aether implied by these experiments provided contradictory results. For example, the Fizeau experiment of 1851 demonstrated that the speed of light in flowing water was less than the speed of light in air plus the speed of the flowing water, the partial aether-dragging implied by this result was in conflict with measurements of stellar aberration. By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein were to derive later, but with a fundamentally different interpretation. As a theory of dynamics, his theory assumed actual physical deformations of the constituents of matter. For example, most physicists believed that Lorentz contraction would be detectable by such experiments as the Trouton–Noble experiment or the Experiments of Rayleigh and Brace. However, these negative results, and in his 1904 theory of the electron. Einstein performed his analyses in terms of kinematics rather than dynamics and it would appear that he did not at first think geometrically about spacetime. It was Einsteins former mathematics professor, Hermann Minkowski, who was to provide an interpretation of special relativity. Einstein was initially dismissive of the interpretation of special relativity
5.
Minkowski spacetime
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Minkowski space is closely associated with Einsteins theory of special relativity, and is the most common mathematical structure on which special relativity is formulated. Because it treats time differently than it treats the three dimensions, Minkowski space differs from four-dimensional Euclidean space. In 3-dimensional Euclidean space, the group is the Euclidean group. It consists of rotations, reflections, and translations, when time is amended as a fourth dimension, the further transformations of translations in time and Galilean boosts are added, and the group of all these transformations is called the Galilean group. All Galilean transformations preserve the 3-dimensional Euclidean distance, Time differences are separately preserved as well. This changes in the spacetime of special relativity, where space, spacetime is equipped with an indefinite non-degenerate bilinear form. Equipped with this product, the mathematical model of spacetime is called Minkowski space. The analogue of the Galilean group for Minkowski space, preserving the interval is the Poincaré group. In summary, Galilean spacetime and Minkowski spacetime are, when viewed as barebones manifolds and they differ in what kind of further structures are defined on them. Here the speed of c is, following Poincare, set to unity. The naming and ordering of coordinates, with the labels for space coordinates. The above expression, while making the expression more familiar. Rotations in planes spanned by two unit vectors appear in coordinate space as well as in physical spacetime appear as Euclidean rotations and are interpreted in the ordinary sense. The analogy with Euclidean rotations is thus only partial and this idea was elaborated by Hermann Minkowski, who used it to restate the Maxwell equations in four dimensions, showing directly their invariance under the Lorentz transformation. He further reformulated in four dimensions the then-recent theory of relativity of Einstein. From this he concluded that time and space should be treated equally, points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point and it is principally this view of spacetime that is current nowadays, although the older view involving imaginary time has also influenced special relativity. An imaginary time coordinate is used also for more subtle reasons in quantum field theory than formal appearance of expressions, in this context, the transformation is called a Wick rotation
6.
Curvature
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In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. This article deals primarily with extrinsic curvature and its canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius everywhere. Smaller circles bend more sharply, and hence have higher curvature, the curvature of a smooth curve is defined as the curvature of its osculating circle at each point. Curvature is normally a scalar quantity, but one may define a curvature vector that takes into account the direction of the bend in addition to its magnitude. The curvature of more objects is described by more complex objects from linear algebra. This article sketches the mathematical framework which describes the curvature of a curve embedded in a plane, the curvature of C at a point is a measure of how sensitive its tangent line is to moving the point to other nearby points. There are a number of equivalent ways that this idea can be made precise and it is natural to define the curvature of a straight line to be constantly zero. The curvature of a circle of radius R should be large if R is small and small if R is large, thus the curvature of a circle is defined to be the reciprocal of the radius, κ =1 R. Given any curve C and a point P on it, there is a circle or line which most closely approximates the curve near P. The curvature of C at P is then defined to be the curvature of that circle or line, the radius of curvature is defined as the reciprocal of the curvature. Another way to understand the curvature is physical, suppose that a particle moves along the curve with unit speed. Taking the time s as the parameter for C, this provides a natural parametrization for the curve, the unit tangent vector T also depends on time. The curvature is then the magnitude of the rate of change of T. Symbolically and this is the magnitude of the acceleration of the particle and the vector dT/ds is the acceleration vector. Geometrically, the curvature κ measures how fast the unit tangent vector to the curve rotates. If a curve close to the same direction, the unit tangent vector changes very little and the curvature is small, where the curve undergoes a tight turn. These two approaches to the curvature are related geometrically by the following observation, in the first definition, the curvature of a circle is equal to the ratio of the angle of an arc to its length. e. For such a curve, there exists a reparametrization with respect to arc length s. This is a parametrization of C such that ∥ γ ′ ∥2 = x ′2 + y ′2 =1, the velocity vector T is the unit tangent vector
7.
Minkowski space
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Minkowski space is closely associated with Einsteins theory of special relativity, and is the most common mathematical structure on which special relativity is formulated. Because it treats time differently than it treats the three dimensions, Minkowski space differs from four-dimensional Euclidean space. In 3-dimensional Euclidean space, the group is the Euclidean group. It consists of rotations, reflections, and translations, when time is amended as a fourth dimension, the further transformations of translations in time and Galilean boosts are added, and the group of all these transformations is called the Galilean group. All Galilean transformations preserve the 3-dimensional Euclidean distance, Time differences are separately preserved as well. This changes in the spacetime of special relativity, where space, spacetime is equipped with an indefinite non-degenerate bilinear form. Equipped with this product, the mathematical model of spacetime is called Minkowski space. The analogue of the Galilean group for Minkowski space, preserving the interval is the Poincaré group. In summary, Galilean spacetime and Minkowski spacetime are, when viewed as barebones manifolds and they differ in what kind of further structures are defined on them. Here the speed of c is, following Poincare, set to unity. The naming and ordering of coordinates, with the labels for space coordinates. The above expression, while making the expression more familiar. Rotations in planes spanned by two unit vectors appear in coordinate space as well as in physical spacetime appear as Euclidean rotations and are interpreted in the ordinary sense. The analogy with Euclidean rotations is thus only partial and this idea was elaborated by Hermann Minkowski, who used it to restate the Maxwell equations in four dimensions, showing directly their invariance under the Lorentz transformation. He further reformulated in four dimensions the then-recent theory of relativity of Einstein. From this he concluded that time and space should be treated equally, points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point and it is principally this view of spacetime that is current nowadays, although the older view involving imaginary time has also influenced special relativity. An imaginary time coordinate is used also for more subtle reasons in quantum field theory than formal appearance of expressions, in this context, the transformation is called a Wick rotation
8.
Smoothness
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In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain, differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives, consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer, the function f is said to be of class Ck if the derivatives f′, f′′. The function f is said to be of class C∞, or smooth, if it has derivatives of all orders. The function f is said to be of class Cω, or analytic, if f is smooth, Cω is thus strictly contained in C∞. Bump functions are examples of functions in C∞ but not in Cω, to put it differently, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous, thus, a C1 function is exactly a function whose derivative exists and is of class C0. In particular, Ck is contained in Ck−1 for every k, C∞, the class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers. The function f = { x if x ≥0,0 if x <0 is continuous, because cos oscillates as x →0, f ’ is not continuous at zero. Therefore, this function is differentiable but not of class C1, the functions f = | x | k +1 where k is even, are continuous and k times differentiable at all x. But at x =0 they are not times differentiable, so they are of class Ck, the exponential function is analytic, so, of class Cω. The trigonometric functions are also analytic wherever they are defined, the function f is an example of a smooth function with compact support. Let n and m be some positive integers, if f is a function from an open subset of Rn with values in Rm, then f has component functions f1. Each of these may or may not have partial derivatives, the classes C∞ and Cω are defined as before. These criteria of differentiability can be applied to the functions of a differential structure. The resulting space is called a Ck manifold, if one wishes to start with a coordinate-independent definition of the class Ck, one may start by considering maps between Banach spaces. A map from one Banach space to another is differentiable at a point if there is a map which approximates it at that point
9.
Curve
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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero, various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows, a curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
10.
Manifold
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of a manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights, two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not, for example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane. When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. One important class of manifolds is the class of differentiable manifolds and this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured, symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. After a line, the circle is the simplest example of a topological manifold, Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 =1. Any point of this arc can be described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the arc to the open interval. Such functions along with the regions they map are called charts. Similarly, there are charts for the bottom, left, and right parts of the circle, together, these parts cover the whole circle and the four charts form an atlas for the circle. The top and right charts, χtop and χright respectively, overlap in their domain, Each map this part into the interval, though differently. Let a be any number in, then, T = χ r i g h t = χ r i g h t =1 − a 2 Such a function is called a transition map. The top, bottom, left, and right charts show that the circle is a manifold, charts need not be geometric projections, and the number of charts is a matter of some choice. These two charts provide a second atlas for the circle, with t =1 s Each chart omits a single point, either for s or for t and it can be proved that it is not possible to cover the full circle with a single chart. Viewed using calculus, the transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable
11.
Tangent space
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The elements of the tangent space are called tangent vectors at x. This is a generalization of the notion of a vector in a Euclidean space. The dimension of all the tangent spaces of a manifold is the same as that of the manifold. More generally, if a manifold is thought of as an embedded submanifold of Euclidean space one can picture a tangent space in this literal fashion. This was the approach to defining parallel transport, and used by Dirac. More strictly this defines a tangent space, distinct from the space of tangent vectors described by modern terminology. In algebraic geometry, in contrast, there is a definition of tangent space at a point P of a variety V. The points P at which the dimension is exactly that of V are called the non-singular points, for example, a curve that crosses itself doesnt have a unique tangent line at that point. The singular points of V are those where the test to be a manifold fails, once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold. A vector field attaches to every point of the manifold a vector from the tangent space at that point, all the tangent spaces can be glued together to form a new differentiable manifold of twice the dimension of the original manifold, called the tangent bundle of the manifold. The informal description above relies on a manifold being embedded in a vector space Rm. However, it is convenient to define the notion of tangent space based on the manifold itself. There are various equivalent ways of defining the tangent spaces of a manifold, while the definition via velocities of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below, in the embedded manifold picture, a tangent vector at a point x is thought of as the velocity of a curve passing through the point x. We can therefore take a tangent vector to be a class of curves passing through x while being tangent to each other at x. Suppose M is a Ck manifold and x is a point in M. Pick a chart φ, U → Rn, where U is an open subset of M containing x. Suppose two curves γ1, → M and γ2, → M with γ1 = γ2 = x are given such that φ ∘ γ1, then γ1 and γ2 are called equivalent at 0 if the ordinary derivatives of φ ∘ γ1 and φ ∘ γ2 at 0 coincide. This defines a relation on such curves, and the equivalence classes are known as the tangent vectors of M at x
12.
Equivalence relation
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In mathematics, an equivalence relation is a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation. As a consequence of these properties an equivalence relation provides a partition of a set into equivalence classes, a given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. That is, for all a, b and c in X, a ~ b if and only if b ~ a. if a ~ b and b ~ c then a ~ c. X together with the relation ~ is called a setoid, the equivalence class of a under ~, denoted, is defined as =. Let the set have the equivalence relation, the following sets are equivalence classes of this relation, =, = =. The set of all classes for this relation is. The following are all equivalence relations, Has the same birthday as on the set of all people, is similar to on the set of all triangles. Is congruent to on the set of all triangles, is congruent to, modulo n on the integers. Has the same image under a function on the elements of the domain of the function, has the same absolute value on the set of real numbers Has the same cosine on the set of all angles. The relation ≥ between real numbers is reflexive and transitive, but not symmetric, for example,7 ≥5 does not imply that 5 ≥7. It is, however, a total order, the relation has a common factor greater than 1 with between natural numbers greater than 1, is reflexive and symmetric, but not transitive. The empty relation R on a non-empty set X is vacuously symmetric and transitive, a partial order is a relation that is reflexive, antisymmetric, and transitive. Equality is both a relation and a partial order. Equality is also the relation on a set that is reflexive. In algebraic expressions, equal variables may be substituted for one another, the equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class. A strict partial order is irreflexive, transitive, and asymmetric, a partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reflexive if and only if for all a ∈ X, a reflexive and symmetric relation is a dependency relation, if finite, and a tolerance relation if infinite. A preorder is reflexive and transitive, a congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure
13.
Equivalence class
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In mathematics, when the elements of some set S have a notion of equivalence defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the equivalence class if. Formally, given a set S and an equivalence relation ~ on S and it may be proven from the defining properties of equivalence relations that the equivalence classes form a partition of S. This partition – the set of equivalence classes – is sometimes called the quotient set or the quotient space of S by ~ and is denoted by S / ~. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories. If X is the set of all cars, and ~ is the relation has the same color as. X/~ could be identified with the set of all car colors. Let X be the set of all rectangles in a plane, for each positive real number A there will be an equivalence class of all the rectangles that have area A. Consider the modulo 2 equivalence relation on the set Z of integers, x ~ y if and this relation gives rise to exactly two equivalence classes, one class consisting of all even numbers, and the other consisting of all odd numbers. Under this relation, and all represent the element of Z/~. Let X be the set of ordered pairs of integers with b not zero, the same construction can be generalized to the field of fractions of any integral domain. In this situation, each equivalence class determines a point at infinity, the equivalence class of an element a is denoted and is defined as the set = of elements that are related to a by ~. An alternative notation R can be used to denote the class of the element a. This is said to be the R-equivalence class of a, the set of all equivalence classes in X with respect to an equivalence relation R is denoted as X/R and called X modulo R. The surjective map x ↦ from X onto X/R, which each element to its equivalence class, is called the canonical surjection or the canonical projection map. When an element is chosen in each class, this defines an injective map called a section. If this section is denoted by s, one has = c for every equivalence class c, the element s is called a representative of c. Any element of a class may be chosen as a representative of the class, sometimes, there is a section that is more natural than the other ones
14.
Arrow of time
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This article is an overview of the subject. For a more technical discussion and for related to current research. The Arrow of Time, or Times Arrow, is a concept developed in 1927 by the British astronomer Arthur Eddington involving the direction or asymmetry of time. This direction, according to Eddington, can be determined by studying the organization of atoms, molecules, and bodies, yet at the macroscopic level it often appears that this is not the case, there is an obvious direction of time. In the 1928 book The Nature of the Physical World, which helped to popularize the concept, Eddington stated and that is the only distinction known to physics. This follows at once if our fundamental contention is admitted that the introduction of randomness is the thing which cannot be undone. I shall use the phrase ‘times arrow’ to express this one-way property of time which has no analogue in space, Eddington then gives three points to note about this arrow, It is vividly recognized by consciousness. It is equally insisted on by our faculty, which tells us that a reversal of the arrow would render the external world nonsensical. It makes no appearance in science except in the study of organization of a number of individuals. According to Eddington the arrow indicates the direction of increase of the random element. Following a lengthy argument upon the nature of thermodynamics he concludes that, so far as physics is concerned, the symmetry of time can be understood by a simple analogy, if time were perfectly symmetrical, a video of real events would seem realistic whether played forwards or backwards. An obvious objection to this notion is gravity, things fall down, yet a ball that is tossed up, slows to a stop and falls into the hand is a case where recordings would look equally realistic forwards and backwards. The system is T-symmetrical but while going forward kinetic energy is dissipated, Entropy may be one of the few processes that is not time-reversible. According to the notion of increasing entropy the arrow of time is identified with a decrease of free energy. If we record somebody dropping a ball that falls for a meter and stops, in reverse we will notice an unrealistic discrepancy, but when the ball lands its kinetic energy is dispersed into sound, shock-waves and heat. In reverse those sound waves, ground vibrations and heat will rush back into the ball, the only unrealism lies in the statistical unlikelihood that such forces could coincide to propel a ball upward into a waiting hand. The arrow of time is the direction or asymmetry of time. The thermodynamic arrow of time is provided by the Second Law of Thermodynamics and this asymmetry can be used empirically to distinguish between future and past though measuring entropy does not accurately measure time
15.
Homeomorphism
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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος = similar and μορφή = shape, roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. A function f, X → Y between two spaces and is called a homeomorphism if it has the following properties, f is a bijection, f is continuous. A function with three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic, a self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form a relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes, the open interval is homeomorphic to the real numbers R for any a < b. The unit 2-disc D2 and the square in R2 are homeomorphic. An example of a mapping from the square to the disc is, in polar coordinates. The graph of a function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve, a chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. The stereographic projection is a homeomorphism between the sphere in R3 with a single point removed and the set of all points in R2. If G is a group, its inversion map x ↦ x −1 is a homeomorphism. Also, for any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, rm and Rn are not homeomorphic for m ≠ n. The Euclidean real line is not homeomorphic to the circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2. The third requirement, that f −1 be continuous, is essential, consider for instance the function f, [0, 2π) → S1 defined by f =
16.
Diffeomorphism
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In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is a function that maps one differentiable manifold to another such that both the function and its inverse are smooth. Given two manifolds M and N, a map f, M → N is called a diffeomorphism if it is a bijection and its inverse f−1. If these functions are r times continuously differentiable, f is called a Cr-diffeomorphism, two manifolds M and N are diffeomorphic if there is a diffeomorphism f from M to N. They are Cr diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable, F is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth. First remark It is essential for V to be connected for the function f to be globally invertible. g. Second remark Since the differential at a point D f x, T x U → T f V is a map, it has a well-defined inverse if. The matrix representation of Dfx is the n × n matrix of partial derivatives whose entry in the i-th row. This so-called Jacobian matrix is used for explicit computations. Third remark Diffeomorphisms are necessarily between manifolds of the same dimension, imagine f going from dimension n to dimension k. If n < k then Dfx could never be surjective, in both cases, therefore, Dfx fails to be a bijection. Fourth remark If Dfx is a bijection at x then f is said to be a local diffeomorphism. Fifth remark Given a smooth map from dimension n to k, if Df is surjective, f is said to be a submersion. Sixth remark A differentiable bijection is not necessarily a diffeomorphism, F = x3, for example, is not a diffeomorphism from R to itself because its derivative vanishes at 0. This is an example of a homeomorphism that is not a diffeomorphism, seventh remark When f is a map between differentiable manifolds, a diffeomorphic f is a stronger condition than a homeomorphic f. For a diffeomorphism, f and its inverse need to be differentiable, for a homeomorphism, f, every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism. F, M → N is called a diffeomorphism if, in coordinate charts, more precisely, Pick any cover of M by compatible coordinate charts and do the same for N. Let φ and ψ be charts on, respectively, M and N, with U and V as, respectively, the map ψfφ−1, U → V is then a diffeomorphism as in the definition above, whenever f ⊂ ψ−1
17.
Monotonic function
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In mathematics, a monotonic function is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was generalized to the more abstract setting of order theory. In calculus, a function f defined on a subset of the numbers with real values is called monotonic if. That is, as per Fig.1, a function that increases monotonically does not exclusively have to increase, a function is called monotonically increasing, if for all x and y such that x ≤ y one has f ≤ f, so f preserves the order. Likewise, a function is called monotonically decreasing if, whenever x ≤ y, then f ≥ f, if the order ≤ in the definition of monotonicity is replaced by the strict order <, then one obtains a stronger requirement. A function with this property is called strictly increasing, again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. The terms non-decreasing and non-increasing should not be confused with the negative qualifications not decreasing, for example, the function of figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing, the term monotonic transformation can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. Notably, this is the case in economics with respect to the properties of a utility function being preserved across a monotonic transform. A function f is said to be absolutely monotonic over an interval if the derivatives of all orders of f are nonnegative or all nonpositive at all points on the interval, F can only have jump discontinuities, f can only have countably many discontinuities in its domain. The discontinuities, however, do not necessarily consist of isolated points and these properties are the reason why monotonic functions are useful in technical work in analysis. In addition, this result cannot be improved to countable, see Cantor function, if f is a monotonic function defined on an interval, then f is Riemann integrable. An important application of functions is in probability theory. If X is a variable, its cumulative distribution function F X = Prob is a monotonically increasing function. A function is unimodal if it is monotonically increasing up to some point, when f is a strictly monotonic function, then f is injective on its domain, and if T is the range of f, then there is an inverse function on T for f. A map f, X → Y is said to be if each of its fibers is connected i. e. for each element y in Y the set f−1 is connected. A subset G of X × X∗ is said to be a set if for every pair. G is said to be monotone if it is maximal among all monotone sets in the sense of set inclusion
18.
Holonomy
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For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features, any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy. The most common forms of holonomy are for connections possessing some kind of symmetry, in each of these cases, the holonomy of the connection can be identified with a Lie group, the holonomy group. The holonomy of a connection is related to the curvature of the connection. The study of Riemannian holonomy has led to a number of important developments, the holonomy was introduced by Cartan in order to study and classify symmetric spaces. It was not until later that holonomy groups would be used to study Riemannian geometry in a more general setting. Later, in 1953, M. Berger classified the possible irreducible holonomies, the decomposition and classification of Riemannian holonomy has applications to physics and to string theory. Let E be a vector bundle over a smooth manifold M. Given a piecewise smooth loop γ, → M based at x in M and this map is both linear and invertible, and so defines an element of the general linear group GL. The holonomy group of ∇ based at x is defined as Hol x =, the restricted holonomy group based at x is the subgroup Hol0x coming from contractible loops γ. If M is connected, then the group depends on the basepoint x only up to conjugation in GL. Explicitly, if γ is a path from x to y in M, choosing different identifications of Ex with Rk also gives conjugate subgroups. Sometimes, particularly in general or informal discussions, one may drop reference to the basepoint, some important properties of the holonomy group include, Hol0 is a connected Lie subgroup of GL. Hol0 is the identity component of Hol, there is a natural, surjective group homomorphism π1 → Hol/Hol0, where π1 is the fundamental group of M, which sends the homotopy class to the coset Pγ·Hol0. If M is simply connected, then Hol = Hol0. ∇ is flat if, the definition for holonomy of connections on principal bundles proceeds in parallel fashion. Let G be a Lie group and P a principal G-bundle over a smooth manifold M which is paracompact, Let ω be a connection on P. Given a piecewise smooth loop γ, → M based at x in M and a point p in the fiber over x, the end point of the horizontal lift, γ ~, will not generally be p but rather some other point p·g in the fiber over x. Define an equivalence relation ~ on P by saying that p ~ q if they can be joined by a smooth horizontal path in P
19.
Interior (topology)
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In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is a point of S. The interior of S is the complement of the closure of the complement of S, in this sense interior and closure are dual notions. The exterior of a set is the interior of its complement, equivalently the complement of its closure, the interior, boundary, and exterior of a subset together partition the whole space into three blocks. The interior and exterior are always open while the boundary is always closed, sets with empty interior have been called boundary sets. If S is a subset of a Euclidean space, then x is a point of S if there exists an open ball centered at x which is completely contained in S. This definition generalizes to any subset S of a metric space X with metric d, x is a point of S if there exists r >0. This definition generalises to topological spaces by replacing open ball with open set, let S be a subset of a topological space X. Then x is a point of S if x is contained in an open subset of S. The interior of a set S is the set of all points of S. The interior of S is denoted int, Int or So, the interior of a set has the following properties. Int is a subset of S. int is the union of all open sets contained in S. int is the largest open set contained in S. A set S is open if and only if S = int. int = int, if S is a subset of T, then int is a subset of int. If A is a set, then A is a subset of S if. Sometimes the second or third property above is taken as the definition of the topological interior, for more on this matter, see interior operator below. In any space, the interior of the empty set is the empty set, in any space X, if A ⊂ X, int is contained in A. If X is the Euclidean space R of real numbers, then int =, if X is the Euclidean space R, then the interior of the set Q of rational numbers is empty. If X is the complex plane C = R2, then i n t =, in any Euclidean space, the interior of any finite set is the empty set
20.
Light cone
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In special and general relativity, a light cone is the path that a flash of light, emanating from a single event and traveling in all directions, would take through spacetime. This view of special relativity was first proposed by Albert Einsteins former professor and is known as Minkowski space, the purpose was to create an invariant spacetime for all observers. To uphold causality, Minkowski restricted spacetime to non-Euclidean hyperbolic geometry, Events which lie neither in the past or future light cone of E cannot influence or be influenced by E in relativity. In special relativity, a cone is the surface describing the temporal evolution of a flash of light in Minkowski spacetime. This can be visualized in 3-space if the two axes are chosen to be spatial dimensions, while the vertical axis is time. The light cone is constructed as follows, given an event E, the light cone classifies all events in spacetime into 5 distinct categories, Events on the future light cone of E. Events on the past light cone of E, Events inside the future light cone of E are those affected by a material particle emitted at E. Events inside the past light cone of E are those that can emit a material particle, all other events are in the elsewhere of E and are those that cannot affect or be affected by E. This is why the concept is so powerful, the above refers to an event occurring at a specific location and at a specific time. To say that one event cannot affect another means that light cannot get from the location of one to the other in an amount of time. Light from each event will make it to the former location of the other. As time progresses, the light cone of a given event will eventually grow to encompass more and more locations. Likewise, if we imagine running time backwards from a given event, the past light cone of an event on present-day Earth, at its very edges, includes very distant objects, but only as they looked long ago, when the Universe was young. Two events at different locations, at the time, are always outside of each others past and future light cones. Since special relativity requires the speed of light to be equal in every inertial frame, commonly a Minkowski diagram is used to illustrate this property of Lorentz transformations. Elsewhere, a part of light cones is the region of spacetime outside the light cone at a given event. Events that are elsewhere from each other are mutually unobservable, in flat spacetime, the future light cone of an event is the boundary of its causal future and its past light cone is the boundary of its causal past. In a curved spacetime, assuming spacetime is globally hyperbolic, it is true that the future light cone of an event includes the boundary of its causal future
21.
Subset
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In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T
22.
Determinism
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Determinism is the philosophical position that for every event there exist conditions that could cause no other event. There are many determinisms, depending on what pre-conditions are considered to be determinative of an event or action, deterministic theories throughout the history of philosophy have sprung from diverse and sometimes overlapping motives and considerations. Some forms of determinism can be tested with ideas from physics. The opposite of determinism is some kind of indeterminism, Determinism is often contrasted with free will. Determinism often is taken to mean causal determinism, which in physics is known as cause-and-effect and it is the concept that events within a given paradigm are bound by causality in such a way that any state is completely determined by prior states. This meaning can be distinguished from varieties of determinism mentioned below. Numerous historical debates involve many philosophical positions and varieties of determinism and they include debates concerning determinism and free will, technically denoted as compatibilistic and incompatibilistic. Determinism should not be confused with self-determination of human actions by reasons, motives, Determinism rarely requires that perfect prediction be practically possible. However, causal determinism is a broad term to consider that ones deliberations, choices. Causal determinism proposes that there is a chain of prior occurrences stretching back to the origin of the universe. The relation between events may not be specified, nor the origin of that universe, causal determinists believe that there is nothing in the universe that is uncaused or self-caused. Historical determinism can also be synonymous with causal determinism, causal determinism has also been considered more generally as the idea that everything that happens or exists is caused by antecedent conditions. Yet they can also be considered metaphysical of origin. Nomological determinism is the most common form of causal determinism and it is the notion that the past and the present dictate the future entirely and necessarily by rigid natural laws, that every occurrence results inevitably from prior events. Quantum mechanics and various interpretations thereof pose a challenge to this view. Nomological determinism is sometimes illustrated by the experiment of Laplaces demon. Nomological determinism is sometimes called scientific determinism, although that is a misnomer, physical determinism is generally used synonymously with nomological determinism. Necessitarianism is closely related to the causal determinism described above and it is a metaphysical principle that denies all mere possibility, there is exactly one way for the world to be. Leucippus claimed there were no uncaused events, and that occurs for a reason
23.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
24.
Conformal map
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In mathematics, a conformal map is a function that preserves angles locally. In the most common case, the function has a domain, more formally, let U and V be subsets of C n. A function f, U → V is called conformal at a point u 0 ∈ U if it preserves oriented angles between curves through u 0 with respect to their orientation. Conformal maps preserve both angles and the shapes of small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation, if the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal. Conformal maps can be defined between domains in higher-dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold, an important family of examples of conformal maps comes from complex analysis. If U is a subset of the complex plane C, then a function f, U → C is conformal if and only if it is holomorphic. If f is antiholomorphic, it preserves angles, but it reverses their orientation. In the literature, there is another definition of conformal maps, since a one-to-one map defined on a non-empty open set cannot be constant, the open mapping theorem forces the inverse function to be holomorphic. Thus, under this definition, a map is conformal if, the two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative, however, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. A map of the complex plane onto itself is conformal if. Again, for the conjugate, angles are preserved, but orientation is reversed, an example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the coordinate in circular coordinates. In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called equivalent if g = u h for some positive function u on M. The function u is called the conformal factor, a diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map, one can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics. If a function is harmonic over a domain, and is transformed via a conformal map to another plane domain
25.
Conformal transformation
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In mathematics, a conformal map is a function that preserves angles locally. In the most common case, the function has a domain, more formally, let U and V be subsets of C n. A function f, U → V is called conformal at a point u 0 ∈ U if it preserves oriented angles between curves through u 0 with respect to their orientation. Conformal maps preserve both angles and the shapes of small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation, if the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal. Conformal maps can be defined between domains in higher-dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold, an important family of examples of conformal maps comes from complex analysis. If U is a subset of the complex plane C, then a function f, U → C is conformal if and only if it is holomorphic. If f is antiholomorphic, it preserves angles, but it reverses their orientation. In the literature, there is another definition of conformal maps, since a one-to-one map defined on a non-empty open set cannot be constant, the open mapping theorem forces the inverse function to be holomorphic. Thus, under this definition, a map is conformal if, the two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative, however, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. A map of the complex plane onto itself is conformal if. Again, for the conjugate, angles are preserved, but orientation is reversed, an example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the coordinate in circular coordinates. In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called equivalent if g = u h for some positive function u on M. The function u is called the conformal factor, a diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map, one can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics. If a function is harmonic over a domain, and is transformed via a conformal map to another plane domain
26.
Causal dynamical triangulation
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This means that it does not assume any pre-existing arena, but rather attempts to show how the spacetime fabric itself evolves. The Loops 05 conference, hosted by many loop quantum gravity theorists, included several presentations which discussed CDT in great depth and it has sparked considerable interest as it appears to have a good semi-classical description. At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d near the Planck scale and these interesting results agree with the findings of Lauscher and Reuter, who use an approach called Quantum Einstein Gravity, and with other recent theoretical work. The same publication gives CDT, and its authors, a feature article in its July 2008 issue. Near the Planck scale, the structure of itself is supposed to be constantly changing due to quantum fluctuations. CDT theory uses a process which varies dynamically and follows deterministic rules. The results of researchers suggest that this is a way to model the early universe. Using a structure called a simplex, it divides spacetime into tiny triangular sections, CDT avoids this problem by allowing only those configurations in which the timelines of all joined edges of simplices agree. CDT is a modification of quantum Regge calculus where spacetime is discretized by approximating it with a piecewise linear manifold in a process called triangulation, in this process, a d-dimensional spacetime is considered as formed by space slices that are labeled by a discrete time variable t. Each space slice is approximated by a simplicial manifold composed by regular -dimensional simplices, in place of a smooth manifold there is a network of triangulation nodes, where space is locally flat but globally curved, as with the individual faces and the overall surface of a geodesic dome. The crucial development is that the network of simplices is constrained to evolve in a way that preserves causality and this allows a path integral to be calculated non-perturbatively, by summation of all possible configurations of the simplices, and correspondingly, of all possible spatial geometries. Simply put, each individual simplex is like a block of spacetime. This rule preserves causality, a feature missing from previous triangulation theories, when simplexes are joined in this way, the complex evolves in an orderly fashion, and eventually creates the observed framework of dimensions. CDT derives the observed nature and properties of spacetime from a set of assumptions. The idea of deriving what is observed from first principles is very attractive to physicists, CDT models the character of spacetime both in the ultra-microscopic realm near the Planck scale, and at the scale of the cosmos, so CDT may provide insights into the nature of reality. Evaluation of the implications of CDT relies heavily on Monte Carlo simulation by computer. Some feel that this makes CDT an inelegant quantum gravity theory, also, it has been argued that discrete time-slicing may not accurately reproduce all possible modes of a dynamical system. However, research by Markopoulou and Smolin demonstrates that the cause for those concerns may be limited, therefore, many physicists still regard this line of reasoning as promising
27.
Closed timelike curve
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In mathematical physics, a closed timelike curve is a world line in a Lorentzian manifold, of a material particle in spacetime that is closed, returning to its starting point. When discussing the evolution of a system in general relativity, or more specifically Minkowski space, a light cone represents any possible future evolution of an object given its current state, or every possible location given its current location. An objects possible future locations are limited by the speed that the object can move, for instance, an object located at position p at time t0 can only move to locations within p + c by time t1. This is commonly represented on a graph with physical locations along the axis and time running vertically, with units of t for time. Light cones in this representation appear as lines at 45 degrees centered on the object, on such a diagram, every possible future location of the object lies within the cone. Additionally, every location has a future time, implying that an object may stay at any location in space indefinitely. Any single point on such a diagram is known as an event, separate events are considered to be timelike if they are separated across the time axis, or spacelike if they differ along the space axis. If the object were in free fall, it would travel up the t-axis, if it accelerates, the actual path an object takes through spacetime, as opposed to the ones it could take, is known as the worldline. Another definition is that the light cone represents all possible worldlines, in simple examples of spacetime metrics the light cone is directed forward in time. This corresponds to the case that an object cannot be in two places at once, or alternately that it cannot move instantly to another location. In these spacetimes, the worldlines of physical objects are, by definition, however this orientation is only true of locally flat spacetimes. In curved spacetimes the light cone will be tilted along the spacetimes geodesic, for instance, while moving in the vicinity of a star, the stars gravity will pull on the object, affecting its worldline, so its possible future positions lie closer to the star. This appears as a slightly tilted lightcone on the spacetime diagram. In extreme examples, in spacetimes with suitably high-curvature metrics, the cone can be tilted beyond 45 degrees. That means there are potential future positions, from the frame of reference. From this outside viewpoint, the object can move instantaneously through space, in these situations the object would have to move, since its present spatial location would not be in its own future light cone. Additionally, with enough of a tilt, there are event locations that lie in the past as seen from the outside, with a suitable movement of what appears to it its own space axis, the object appears to travel though time as seen externally. An object in such an orbit would repeatedly return to the point in spacetime if it stays in free fall
28.
Penrose diagram
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In theoretical physics, a Penrose diagram is a two-dimensional diagram capturing the causal relations between different points in spacetime. It is an extension of a Minkowski diagram where the vertical dimension represents time, and the horizontal dimension represents space, the biggest difference is that locally, the metric on a Penrose diagram is conformally equivalent to the actual metric in spacetime. The conformal factor is such that the entire infinite spacetime is transformed into a Penrose diagram of finite size. For spherically symmetric spacetimes, every point in the diagram corresponds to a 2-sphere, straight lines of constant time and straight lines of constant space ordinates therefore become hyperbolas, which appear to converge at points in the corners of the diagram. These points represent conformal infinity for space and time, Penrose diagrams are more properly called Penrose–Carter diagrams, acknowledging both Brandon Carter and Roger Penrose, who were the first researchers to employ them. They are also called conformal diagrams, or simply spacetime diagrams, two lines drawn at 45° angles should intersect in the diagram only if the corresponding two light rays intersect in the actual spacetime. So, a Penrose diagram can be used as an illustration of spacetime regions that are accessible to observation. The diagonal boundary lines of a Penrose diagram correspond to the infinity or to singularities where light rays must end, thus, Penrose diagrams are also useful in the study of asymptotic properties of spacetimes and singularities. Penrose diagrams are used to illustrate the causal structure of spacetimes containing black holes. Singularities are denoted by a boundary, unlike the timelike boundary found on conventional space-time diagrams. This is due to the interchanging of timelike and spacelike coordinates within the horizon of a black hole. The singularity is represented by a boundary to make it clear that once an object has passed the horizon it will inevitably hit the singularity even if it attempts to take evasive action. Penrose diagrams are used to illustrate the hypothetical Einstein-Rosen bridge connecting two separate universes in the maximally extended Schwarzschild black hole solution. The precursors to the Penrose diagrams were Kruskal–Szekeres diagrams and these introduced the method of aligning the event horizon into past and future horizons oriented at 45° angles, and splitting the singularity into past and future horizontally-oriented lines. The Einstein-Rosen bridge closes off so rapidly that passage between the two asymptotically flat exterior regions would require faster-than-light velocity, and is therefore impossible, in addition, highly blue-shifted light rays would make it impossible for anyone to pass through. In the case of the hole, there is also a negative universe entered through a ring-shaped singularity that can be passed through if entering the hole close to its axis of rotation. Causality Causal structure Conformal cyclic cosmology Weyl transformation dInverno, Ray, see Chapter 17 for a very readable introduction to the concept of conformal infinity plus examples. Complete Analytic Extension of the Symmetry Axis of Kerrs Solution of Einsteins Equations, see also on-line version Hawking, Stephen & Ellis, G. F. R
29.
Stephen Hawking
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Hawking was the first to set forth a theory of cosmology explained by a union of the general theory of relativity and quantum mechanics. He is a supporter of the many-worlds interpretation of quantum mechanics. In 2002, Hawking was ranked number 25 in the BBCs poll of the 100 Greatest Britons, Hawking has a rare early-onset, slow-progressing form of amyotrophic lateral sclerosis that has gradually paralysed him over the decades. He now communicates using a single cheek muscle attached to a speech-generating device, Hawking was born on 8 January 1942 in Oxford, England to Frank and Isobel Hawking. Despite their families financial constraints, both attended the University of Oxford, where Frank read medicine and Isobel read Philosophy. The two met shortly after the beginning of the Second World War at a research institute where Isobel was working as a secretary. They lived in Highgate, but, as London was being bombed in those years, Hawking has two younger sisters, Philippa and Mary, and an adopted brother, Edward. In 1950, when Hawkings father became head of the division of parasitology at the National Institute for Medical Research, Hawking and his moved to St Albans. In St Albans, the family were considered intelligent and somewhat eccentric. They lived an existence in a large, cluttered, and poorly maintained house. During one of Hawkings fathers frequent absences working in Africa, the rest of the family spent four months in Majorca visiting his mothers friend Beryl and her husband, Hawking began his schooling at the Byron House School in Highgate, London. He later blamed its progressive methods for his failure to learn to read while at the school, in St Albans, the eight-year-old Hawking attended St Albans High School for Girls for a few months. At that time, younger boys could attend one of the houses, the family placed a high value on education. Hawkings father wanted his son to attend the well-regarded Westminster School and his family could not afford the school fees without the financial aid of a scholarship, so Hawking remained at St Albans. From 1958 on, with the help of the mathematics teacher Dikran Tahta, they built a computer from clock parts, although at school Hawking was known as Einstein, Hawking was not initially successful academically. With time, he began to show aptitude for scientific subjects and, inspired by Tahta. Hawkings father advised him to medicine, concerned that there were few jobs for mathematics graduates. He wanted Hawking to attend University College, Oxford, his own alma mater, as it was not possible to read mathematics there at the time, Hawking decided to study physics and chemistry
30.
George F. R. Ellis
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He co-authored The Large Scale Structure of Space-Time with University of Cambridge physicist Stephen Hawking, published in 1973, and is considered one of the worlds leading theorists in cosmology. He is an active Quaker and in 2004 he won the Templeton Prize, from 1989 to 1992 he served as President of the International Society on General Relativity and Gravitation. He is a past President of the International Society for Science and he is an A-rated researcher with the NRF. He was also awarded the Order of the Star of South Africa by Nelson Mandela, on 18 May 2007, he was elected a Fellow of the British Royal Society. In 2005 Ellis appeared as a guest speaker at the Nobel Conference in St. Peter and he represented the university in fencing, rowing and flying. While a student at Cambridge University, where he received a PhD. in applied maths and theoretical physics in 1964, the following year, Ellis co-wrote The Large Scale Structure of Space-Time with Stephen Hawking, debuting at a strategic moment in the development of General Relativity Theory. In the following year, Ellis returned to South Africa to accept an appointment as Professor of Applied Mathematics at the University of Cape Town, george Ellis has worked for many decades on anisotropic cosmologies and inhomogeneous universes, and on the philosophy of cosmology. He is currently writing on the emergence of complexity, and the way this is enabled by top-down causation in the hierarchy of complexity, in terms of philosophy of science, Ellis is a Platonist. The Large Scale Structure of Space-Time, low Income Housing Policy in South Africa, Urban Problems Research Unit, UCT,1979. Flat and Curved Space Times, Oxford University Press,1988, before the Beginning, Cosmology Explained, Bowerdean/Marion Boyars,1993. The Renaissance of General Relativity and Cosmology, University Press, Cambridge 1993, paperback,2005. Science Research Policy in South Africa, Royal Society of South Africa,1994, on The Moral Nature of the universe, Cosmology, Theology, and Ethics. Is The Universe Open or Closed, the Density of Matter in the Universe. The Far Future Universe, Templeton Foundation Press,2002, Science in Faith and Hope, an interaction, Quaker Books,2004. Relativistic Cosmology, Cambridge University Press,2012, how Can Physics Underlie the Mind. Top-Down Causation in the Human Context, Springer,2016 Ellis has over 500 published articles including 17 in Nature
31.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
32.
Roger Penrose
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Sir Roger Penrose OM FRS is an English mathematical physicist, mathematician and philosopher of science. He is the Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute of the University of Oxford, Penrose is known for his work in mathematical physics, in particular for his contributions to general relativity and cosmology. He has received prizes and awards, including the 1988 Wolf Prize for physics. Penrose told a Russian audience that his grandmother had left St. Petersburg in the late 1880s and his uncle was artist Roland Penrose, whose son with photographer Lee Miller is Antony Penrose. Penrose is the brother of physicist Oliver Penrose and of chess Grandmaster Jonathan Penrose, Penrose attended University College School and University College, London, where he graduated with a first class degree in mathematics. In 1955, while still a student, Penrose reintroduced the E. H. Moore generalised matrix inverse, also known as the Moore–Penrose inverse, after it had been reinvented by Arne Bjerhammar in 1951. He devised and popularised the Penrose triangle in the 1950s, describing it as impossibility in its purest form, Escher, whose earlier depictions of impossible objects partly inspired it. Eschers Waterfall, and Ascending and Descending were in inspired by Penrose. As reviewer Manjit Kumar puts it, As a student in 1954, soon he was trying to conjure up impossible figures of his own and discovered the tribar – a triangle that looks like a real, solid three-dimensional object, but isnt. Together with his father, a physicist and mathematician, Penrose went on to design a staircase that simultaneously loops up, an article followed and a copy was sent to Escher. Completing a cyclical flow of creativity, the Dutch master of illusions was inspired to produce his two masterpieces. One approach to issue was by the use of perturbation theory. The importance of Penroses epoch-making paper Gravitational collapse and space-time singularities was not only its result, following up his weak cosmic censorship hypothesis, Penrose went on, in 1979, to formulate a stronger version called the strong censorship hypothesis. Together with the BKL conjecture and issues of stability, settling the censorship conjectures is one of the most important outstanding problems in general relativity. Also from 1979 dates Penroses influential Weyl curvature hypothesis on the conditions of the observable part of the universe. Penrose and James Terrell independently realised that objects travelling near the speed of light appear to undergo a peculiar skewing or rotation. This effect has come to be called the Terrell rotation or Penrose–Terrell rotation, in 1967, Penrose invented the twistor theory which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature. Penrose developed these ideas based on the article Deux types fondamentaux de distribution statistique by Czech geographer, demographer, in 1984, such patterns were observed in the arrangement of atoms in quasicrystals
33.
Gary Gibbons
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Gary William Gibbons FRS is a British theoretical physicist. Gibbons was born in Coulsdon, Surrey and he was educated at Purley County Grammar School and the University of Cambridge, where in 1969 he became a research student under the supervision of Dennis Sciama. When Sciama moved to the University of Oxford, he became a student of Stephen Hawking, obtaining his PhD from Cambridge in 1973, having worked on classical general relativity for his PhD thesis, Gibbons focused on the quantum theory of black holes afterwards. Together with Malcolm Perry, he used thermal Greens functions to prove the universality of thermodynamic properties of horizons and he developed the Euclidean approach to quantum gravity with Stephen Hawking, which allows a derivation of the thermodynamics of black holes from a functional integral approach. As the Euclidean action for gravity is not positive definite, the integral only converges when a particular contour is used for conformal factors and his work in more recent years includes contributions to research on supergravity, p-branes and M-theory, mainly motivated by string theory. Gibbons remains interested in problems of all sorts which have applications to physics. Gibbons was elected a Fellow of the Royal Society in 1999